Ch. 10 - Chi-Square Tests and the F-Distribution

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 10 - Chi-Square Tests and the F-Distribution

Problem 10.1.10a

Chapter 10, Problem 10.1.10a

Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ.

Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)

Verified step by step guidance

Step 1: Identify the claim and hypotheses. The claim is that the distribution of people's preferences for payment methods is different from the given distribution. Define the null hypothesis (H₀) as 'The distribution of preferences matches the given distribution' and the alternative hypothesis (Hₐ) as 'The distribution of preferences is different from the given distribution.'

Step 2: Calculate the expected frequencies for each payment method based on the given percentages and the total sample size of 600 people. Use the formula: Expected frequency = (Percentage / 100) × Total sample size. For example, for 'Cash,' the expected frequency is (29 / 100) × 600.

Step 3: Compute the Chi-Square test statistic using the formula: χ² = Σ((Observed frequency - Expected frequency)² / Expected frequency). For each payment method, subtract the expected frequency from the observed frequency, square the result, divide by the expected frequency, and sum these values across all categories.

Step 4: Determine the degrees of freedom (df) for the test. The degrees of freedom are calculated as df = (Number of categories - 1). In this case, there are 4 categories (Cash, Debit or credit, Check, Digital wallet/other), so df = 4 - 1 = 3.

Step 5: Compare the calculated χ² value to the critical value from the Chi-Square distribution table at α = 0.01 and df = 3. If the χ² value exceeds the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject H₀.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chi-Square Goodness-of-Fit Test

The Chi-Square Goodness-of-Fit Test is a statistical method used to determine if the observed frequencies of a categorical variable differ significantly from the expected frequencies based on a specific hypothesis. It compares the actual data collected from a sample to a theoretical distribution, allowing researchers to assess whether the sample data fits the expected distribution.

Null and Alternative Hypotheses (H₀ and Hₐ)

In hypothesis testing, the null hypothesis (H₀) represents a statement of no effect or no difference, suggesting that any observed variation is due to chance. The alternative hypothesis (Hₐ) posits that there is a significant effect or difference. In this context, H₀ would state that the distribution of payment preferences matches the expected distribution, while Hₐ would claim that it differs.

Significance Level (α)

The significance level (α) is a threshold set by the researcher to determine the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. In this case, α = 0.01 indicates a 1% risk of concluding that a difference exists when there is none. This level is used to assess the strength of the evidence against the null hypothesis in the context of the Chi-Square test.

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Related Practice

Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (b) find the critical value and identify the rejection region.

Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)

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Textbook Question

In each exercise,

e. interpret the decision in the context of the original claim.

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

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Textbook Question

In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.05,d.f.N=20,d.f.D=25

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Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (c) find the chi-square test statistic.

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Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (d) decide whether to reject or fail to reject the null hypothesis.

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Textbook Question

In each exercise,

d. decide whether to reject or fail to reject the null hypothesis, and

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

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